Finite Group Databasebeta

← #27302430

Subgroups — #27302430

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276390C211yesmaximal, characteristic, complemented
2#20276387C211yesmaximal, characteristic
3#20276369C211yesmaximal, characteristic, complemented
4#20276365C211yesmaximal, characteristic
5#20274761C211yesmaximal, characteristic, complemented
6#20274515C211yesmaximal, characteristic, complemented
7C53×#3057C211yesmaximal, characteristic
8#16810661—21no—
9#16810607C2211yescharacteristic
10#16810597C2211yescharacteristic
11#16810597C2211yescharacteristic
12#16810597C2211yescharacteristic
13#16810578—21no—
14#16810418—22no—
15#16810416—22no—
16#16810413—22no—
17#16810410—22no—
18#16810394—21no—
19#16810380—21no—
20C22×SD16×C53C411yescharacteristic, complemented
21C2×#131×C53C2211yescharacteristic
22C53×#536C2211yescharacteristic
23C53×#535C2211yescharacteristic
24C53×#468—21no—
25C53×#468—21no—
26C53×#465—21no—
27C53×#465—21no—
28C53×#427C411yescharacteristic, complemented
29C53×#419C411yescharacteristic, complemented
30C53×#418C411yescharacteristic, complemented
31C2×#11719917—21no—
32#13133176—21no—
33#13133176—21no—
34#13133176—21no—
35#13133151—41no—
36#13133149—41no—
37#13133136—21no—
38#13133134D412yes—
39#13133132—21no—
40#13133132—21no—
41#13133132—21no—
42#13133132—21no—
43#13133132—21no—
44#13133132D412yes—
45#13133127—41no—
46C8×Dic53—41no—
47C2×#41×C53C2311yescharacteristic
48C22×Q8×C53C2×C411yescharacteristic
49C22×D4×C53C2×C411yescharacteristic
50C2×SD16×C53—21no—
51C2×SD16×C53—21no—
52C2×SD16×C53—41no—
53C2×SD16×C53—41no—
54#131×C53—41no—
55#131×C53—41no—
56C2×M16×C53C2×C411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C2×M16×C53D411yescharacteristic
59C22×C8×C53C2×C411yescharacteristic
60C22×C8×C53D411yescharacteristic
61C22×C8×C53D411yescharacteristic
62C4⋊Q8×C53—21no—
63#127×C53—21no—
64#124×C53—21no—
65#124×C53—21no—
66#123×C53—21no—
67#123×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72#120×C53—21no—
73#120×C53—21no—
74#117×C53C2×C411yescharacteristic
75#117×C53C2×C411yescharacteristic
76#107×C53—21no—
77#106×C53—21no—
78#103×C53—21no—
79#103×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82C24×C53—21no—
83#41×C53—41no—
84#41×C53—41no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×Q8×C53#4111yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53C2×D411yescharacteristic
97C2×D4×C53C2×D411yescharacteristic
98C2×D4×C53#4111yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103SD16×C53—42no—
104SD16×C53—41no—
105SD16×C53—41no—
106M16×C53—41no—
107M16×C53—41no—
108M16×C53—41no—
109C2×C8×C53—21no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53C22⋊C411yescharacteristic
116C2×C8×C53C22⋊C411yescharacteristic
117C2×C8×C53C22⋊C411yescharacteristic
118C2×C8×C53C22⋊C411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C42×C53—21no—
130C42×C53—21no—
131C22⋊Dic53—22no—
132C22⋊Dic53—41no—
133C22⋊Dic53—41no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C4×Dic53—41no—
140#11719917—22no—
141#11719917—41no—
142C2×#88322—41no—
143C23×C53—41no—
144C23×C53—81no—
145C23×C53—161no—
146C23×C53C4×D411yescharacteristic
147C23×C53#12111yescharacteristic
148C23×C53#12711yescharacteristic
149Q8×C53—41no—
150Q8×C53—41no—
151Q8×C53—41no—
152Q8×C53—81no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—81no—
159D4×C53—161no—
160D4×C53—161no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—81no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12411yescharacteristic
175C2×Dic53—42no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C8×C53—21no—
179C8×C53—21no—
180C8×C53—21no—
181C8×C53#9911yescharacteristic
182C8×C53#9911yescharacteristic
183#88322—42no—
184#13229—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—162no—
195C22×C53#39011yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53C2×#9911yescharacteristic
203C4×C53#41011yescharacteristic
204Dic53—82no—
205Dic53—161no—
206Dic53—161no—
207#3057D5311yescharacteristic
208#2129—531no—
209#2128—531no—
210#2123—531no—
211#2122—531no—
212#1812—531no—
213#1757—531no—
214C2×C53—21no—
215C2×C53—41no—
216C2×C53—41no—
217C2×C53—41no—
218C2×C53—161no—
219C2×C53#196411yescharacteristic
220C22×SD16Dic5311yescharacteristic, complemented
221C2×#131C2×D5311yescharacteristic
222#536C2×D5311yescharacteristic
223#535C2×D5311yescharacteristic
224#468—21no—
225#468—21no—
226#465—21no—
227#465—21no—
228#427—1061no—
229#427Dic5311yescharacteristic, complemented
230#419Dic5311yescharacteristic, complemented
231#418Dic5311yescharacteristic, complemented
232#413—531no—
233#410—531no—
234#410—531no—
235#410—531no—
236#405—1061no—
237#354—1062no—
238#353—1062no—
239#352—1062no—
240AΣL(1,𝔽16)—1062no—
241#344—1061no—
242H(3,ℤ/4)—1061no—
243C53#1322911yescharacteristic, complemented, Sylow
244C2×#41C22×D5311yescharacteristic
245C22×Q8C2×Dic5311yescharacteristic
246C22×D4C2×Dic5311yescharacteristic
247C2×SD16—21no—
248C2×SD16—21no—
249C2×SD16—41no—
250C2×SD16—41no—
251#131—41no—
252#131—41no—
253C2×M16—1061no—
254C2×M16C2×Dic5311yescharacteristic
255C2×M16C22⋊D5311yescharacteristic
256C2×M16C22⋊D5311yescharacteristic
257C22×C8C2×Dic5311yescharacteristic
258C22×C8C22⋊D5311yescharacteristic
259C22×C8C22⋊D5311yescharacteristic
260C4⋊Q8—21no—
261#127—21no—
262#124—21no—
263#124—21no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#120—21no—
271#120—21no—
272#117—1061no—
273#117—1061no—
274#117—1061no—
275#117C2×Dic5311yescharacteristic
276#117C2×Dic5311yescharacteristic
277#107—21no—
278#107—2121no—
279#106—21no—
280#106—2121no—
281#103—21no—
282#103—21no—
283#102—21no—
284#102—21no—
285#101—1061no—
286#100—532no—
287#99—532no—
288#99—1061no—
289#99—1061no—
290#99—1061no—
291#99—1061no—
292#99—1061no—
293#97—2121no—
294C4×C8—2121no—
295C24—21no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8#1171994711yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4D4×D5311yescharacteristic
310C2×D4D4×D5311yescharacteristic
311C2×D4#1171994711yescharacteristic
312C22×C4—21no—
313C22×C4C22×Dic5311yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316SD16—42no—
317SD16—41no—
318SD16—41no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—1062no—
323M16—2121no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—2121no—
331C2×C8C22⋊Dic5311yescharacteristic
332C2×C8C22⋊Dic5311yescharacteristic
333C2×C8C22⋊Dic5311yescharacteristic
334C2×C8C22⋊Dic5311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—1062no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C42—21no—
352C42—21no—
353C42—2121no—
354C42—2121no—
355C42—2121no—
356C23—41no—
357C23—81no—
358C23—161no—
359C23D4×Dic5311yescharacteristic
360C23#1313318811yescharacteristic
361C23#1313321311yescharacteristic
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—81no—
372D4—161no—
373D4—161no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—2122no—
387C2×C4C2×C22⋊Dic5311yescharacteristic
388C2×C4D4×Dic5311yescharacteristic
389C2×C4#1313318811yescharacteristic
390C2×C4#1313320411yescharacteristic
391C8—21no—
392C8—21no—
393C8—21no—
394C8—2122no—
395C8#1313313211yescharacteristic
396C8#1313313211yescharacteristic
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—162no—
407C22#1681052611yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—8481no—
415C4—8481no—
416C4—4242no—
417C4C2×#1313313211yescharacteristic
418C4#1681059711yescharacteristic
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—161no—
424C2#2027555211yescharacteristic
subgroupG/H
centerC2#20275552
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3057C2
socleC2×C53#1964
prime radicalC53×#535C22
cosocleC2×#41×C53C23
hypercenter#3057D53
hypocenterC53#13229
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #3057

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20276390C2semidirect
3. #20276369C2semidirect
5. #20274761C2semidirect
6. #20274515C2semidirect
20. C22×SD16×C53C4semidirect
28. C53×#427C4semidirect
29. C53×#419C4semidirect
30. C53×#418C4semidirect
220. C22×SD16Dic53semidirect
229. #427Dic53semidirect
230. #419Dic53semidirect
231. #418Dic53semidirect
243. C53#13229semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202763902
#202763872
#202763692
#202763652
#202747612
#202745152
C53×#30572
#1322953